Young’s Modulus
For a solid, in the form of a wire or a thin rod, Young’s modulus of elasticity within elastic limit is defined as the ratio of longitudinal stress to longitudinal strain. It is given as:
It has the unit of longitudinal stress and dimensions of [MI. ^{1}T^{2}] Its unit is Pascal or N/m^{2}.
Bulk Modulus
Within elastic limit the bulk modulus is defined as the ratio of longitudinal stress and volumetric strain. It is given as:
– ve indicates that the volume variation and pressure variation always negate each other.
It is defined as the ratio of the tangential stress to the shear strain.
Modulus of rigidity is given by
The ratio of change in diameter (ΔD) to the original diameter (D) is called lateral strain. The ratio of change in length (Δl) to the original length (l) is called longitudinal strain. The ratio of lateral strain to the longitudinal strain is called Poisson’s ratio.
For most of the substances, the value of σ lies between 0.2 to 0.4 When a boay is perfectly incompressible, the value of σ is maximum and equals to 0.5.
It is the property of an elastic body by virtue of which its behaviour becomes less elastic under the action of repeated alternating deforming forces.
For isotropic materials (i.e., materials having the same properties in all directions), only two of the three elastic constants are independent. For example, Young’s modulus can be expressed in terms of the bulk and shear moduli.
The ultimate tensile strength of a material is the stress required to break a wire or a rod by pulling on it. The breaking stress of the material is the maximum stress which a material can withstand. Beyond this point breakage occurs.
Work done to stretch wire
According to the formula given by
Where F is the force needed to stretch the wire of length L and area of crosssection A, is the increase in the length of the wire.
The work done by this force in stretching the wire is stored in the wire as potential energy.
Integrating both sides, we get
Which equal to the elastic potential energy U.
Now the potential energy per unit volume is
Hence, the elastic potential energy of a wire (energy density) is equal to half the product of its stress and strain.
TABLE 9.1 Young's modulus, elastic limit and tensile strengths of some materials.
Substance  Young's modulus 10^{9}N/m^{2} σ_{y}  Elastic limit 10^{7}N/m^{2} %  Tensile strength 10^{7}N/m^{2} σ_{u} 
Aluminium  70  IS  20 
Copper  120  20  40 
Iron {wrought)  190  17  33 
Steel  200  30  50 
Bone 



(Tensile)  16 
 12 
(Compressive)  9 
 12 
TABLE 9.2 Shear modulus (G) of some common materials
Material  G(10^{9} Nm^{2} or GPa) 
Aluminium  25 
Brass  36 
Copper  42 
Glass  23 
Iron  70 
Lead  5.6 
Nickel  77 
Steel  84 
Tungsten  150 
Wood  10 
TABLE 9.3 Bulk modulus (B) of some common Materials
Material Solids  B(10^{9} Nm^{2} or GPa) 
Aluminium  72 
Brass  61 
Copper  140 
Glass  37 
Iron  100 
Nickel  260 
Steel  160 
Liquids 

Water  2.2 
Ethanol  0.9 
Carbon disulphide  1.56 
Glycerine  4.76 
Mercury  25 
Gases 

Air (at 5TP)  1.0 * 10^{4} 
TABLE 9.4 Stress, strain and various elastic modulus
Type of Stress  Stress  Strain  Change in  Elastic modulus  Name of  State of Mater  
shape  volume  
Tensile or Compressive  Two equal and opposite forces perpendicular to Opposite faces (σ = F/A)  Elongation or compression parallel to force direction (ΔL/L) (longitudinal strain)  Yes  No  V = (FxL) /(A » ΔL)  Young's modulus  Solid 
Shearing  Two equal and opposite forces parallel to opposite surfaces [forces in each case such that total force and total torque on the body vanishes (σ_{s} * F/A)]  Pure shear, θ  Yes  No  C  (F x θ) /A  Shear modulus  Solid 
Hydraulic  Forces perpendicular every where to the surface, force per unit area (pressure) same everywhere  Volume change (compression or elongation) (ΔV/V)  No  Yes  B =  p/(Δ V)  Bulk modulus  Solid, liquid and gas 